Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada$$\frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x^{2}} - \frac{2 \sin^{2}{\left(x \right)}}{x^{3}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 65.9734457253857$$
$$x_{2} = 69.1150383789755$$
$$x_{3} = 45.5311340139913$$
$$x_{4} = -21.9911485751286$$
$$x_{5} = 21.9911485751286$$
$$x_{6} = 14.0661939128315$$
$$x_{7} = 58.1022547544956$$
$$x_{8} = -11295.596297452$$
$$x_{9} = -3.14159265358979$$
$$x_{10} = -15.707963267949$$
$$x_{11} = -20.3713029592876$$
$$x_{12} = -86.3822220347287$$
$$x_{13} = -89.5242209304172$$
$$x_{14} = -45.5311340139913$$
$$x_{15} = -80.0981286289451$$
$$x_{16} = 6.28318530717959$$
$$x_{17} = 95.8081387868617$$
$$x_{18} = 73.8138806006806$$
$$x_{19} = -58.1022547544956$$
$$x_{20} = -42.3879135681319$$
$$x_{21} = 29.811598790893$$
$$x_{22} = 28.2743338823081$$
$$x_{23} = -94.2477796076938$$
$$x_{24} = -12.5663706143592$$
$$x_{25} = -73.8138806006806$$
$$x_{26} = 51.8169824872797$$
$$x_{27} = -67.5294347771441$$
$$x_{28} = 36.1006222443756$$
$$x_{29} = -53.4070751110265$$
$$x_{30} = -4.49340945790906$$
$$x_{31} = 42.3879135681319$$
$$x_{32} = -95.8081387868617$$
$$x_{33} = 67.5294347771441$$
$$x_{34} = 185.353966561798$$
$$x_{35} = 59.6902604182061$$
$$x_{36} = 56.5486677646163$$
$$x_{37} = -36.1006222443756$$
$$x_{38} = 72.2566310325652$$
$$x_{39} = -50.2654824574367$$
$$x_{40} = 92.6661922776228$$
$$x_{41} = 78.5398163397448$$
$$x_{42} = 48.6741442319544$$
$$x_{43} = -87.9645943005142$$
$$x_{44} = 37.6991118430775$$
$$x_{45} = -6.28318530717959$$
$$x_{46} = -7.72525183693771$$
$$x_{47} = -37.6991118430775$$
$$x_{48} = -43.9822971502571$$
$$x_{49} = 47.1238898038469$$
$$x_{50} = -100.530964914873$$
$$x_{51} = 3.14159265358979$$
$$x_{52} = -72.2566310325652$$
$$x_{53} = -81.6814089933346$$
$$x_{54} = 26.6660542588127$$
$$x_{55} = -65.9734457253857$$
$$x_{56} = -28.2743338823081$$
$$x_{57} = -56.5486677646163$$
$$x_{58} = -207.345115136926$$
$$x_{59} = -54.9596782878889$$
$$x_{60} = 80.0981286289451$$
$$x_{61} = -39.2444323611642$$
$$x_{62} = -83.2401924707234$$
$$x_{63} = 43.9822971502571$$
$$x_{64} = -61.2447302603744$$
$$x_{65} = 100.530964914873$$
$$x_{66} = -97.3893722612836$$
$$x_{67} = 81.6814089933346$$
$$x_{68} = -75.398223686155$$
$$x_{69} = 50.2654824574367$$
$$x_{70} = 94.2477796076938$$
$$x_{71} = 64.3871195905574$$
$$x_{72} = -59.6902604182061$$
$$x_{73} = 12.5663706143592$$
$$x_{74} = -64.3871195905574$$
$$x_{75} = 23.519452498689$$
$$x_{76} = -23.519452498689$$
$$x_{77} = 86.3822220347287$$
$$x_{78} = -939.336203423348$$
$$x_{79} = -51.8169824872797$$
$$x_{80} = 34.5575191894877$$
$$x_{81} = 15.707963267949$$
$$x_{82} = 87.9645943005142$$
$$x_{83} = -14.0661939128315$$
$$x_{84} = -9.42477796076938$$
$$x_{85} = 4.49340945790906$$
$$x_{86} = 20.3713029592876$$
$$x_{87} = -29.811598790893$$
$$x_{88} = 89.5242209304172$$
$$x_{89} = 70.6716857116195$$
$$x_{90} = -31.4159265358979$$
$$x_{91} = 7.72525183693771$$
Signos de extremos en los puntos:
(65.97344572538566, 2.21085464398688e-34)
(69.11503837897546, 4.07351440822617e-33)
(45.53113401399128, 0.0004821405114931)
(-21.991148575128552, 1.51957436358475e-33)
(21.991148575128552, 1.51957436358475e-33)
(14.066193912831473, 0.00502871873123234)
(58.10225475449559, 0.000296132041061176)
(-11295.596297452026, 7.83757431584905e-9)
(-3.141592653589793, 1.51957436358475e-33)
(-15.707963267948966, 1.51957436358475e-33)
(-20.37130295928756, 0.00240390403096148)
(-86.38222203472871, 0.000133996378076552)
(-89.52422093041719, 0.000124756940214054)
(-45.53113401399128, 0.0004821405114931)
(-80.09812862894512, 0.000155843098300362)
(6.283185307179586, 1.51957436358475e-33)
(95.8081387868617, 0.00010893009510268)
(73.81388060068065, 0.000183503445117105)
(-58.10225475449559, 0.000296132041061176)
(-42.38791356813192, 0.000556255443367358)
(29.81159879089296, 0.00112393467820302)
(28.274333882308138, 1.51957436358475e-33)
(-94.2477796076938, 1.32563038769384e-33)
(-12.566370614359172, 1.51957436358475e-33)
(-73.81388060068065, 0.000183503445117105)
(51.81698248727967, 0.000372300864235917)
(-67.52943477714412, 0.000219239370163893)
(36.10062224437561, 0.000766721274909305)
(-53.40707511102649, 7.58434347792408e-34)
(-4.493409457909064, 0.0471904492258113)
(42.38791356813192, 0.000556255443367358)
(-95.8081387868617, 0.00010893009510268)
(67.52943477714412, 0.000219239370163893)
(185.3539665617978, 3.38129490222314e-33)
(59.69026041820607, 4.21786179228739e-34)
(56.548667764616276, 1.51957436358475e-33)
(-36.10062224437561, 0.000766721274909305)
(72.25663103256524, 7.77037197267108e-33)
(-50.26548245743669, 1.51957436358475e-33)
(92.66619227762284, 0.000116441231903146)
(78.53981633974483, 3.90979723557589e-35)
(48.674144231954386, 0.000421910252241397)
(-87.96459430051421, 1.51957436358475e-33)
(37.69911184307752, 1.51957436358475e-33)
(-6.283185307179586, 1.51957436358475e-33)
(-7.725251836937707, 0.0164800259929739)
(-37.69911184307752, 1.51957436358475e-33)
(-43.982297150257104, 1.51957436358475e-33)
(47.1238898038469, 1.32563038769384e-33)
(-100.53096491487338, 1.51957436358475e-33)
(3.141592653589793, 1.51957436358475e-33)
(-72.25663103256524, 7.77037197267108e-33)
(-81.68140899333463, 2.30474995774498e-33)
(26.666054258812675, 0.00140433964877555)
(-65.97344572538566, 2.21085464398688e-34)
(-28.274333882308138, 1.51957436358475e-33)
(-56.548667764616276, 1.51957436358475e-33)
(-207.34511513692635, 2.22134595698259e-35)
(-54.959678287888934, 0.000330954187793896)
(80.09812862894512, 0.000155843098300362)
(-39.24443236116419, 0.000648876433872722)
(-83.2401924707234, 0.000144301609334975)
(43.982297150257104, 1.51957436358475e-33)
(-61.2447302603744, 0.000266530417407147)
(100.53096491487338, 1.51957436358475e-33)
(-97.3893722612836, 4.96414972110828e-33)
(81.68140899333463, 2.30474995774498e-33)
(-75.39822368615503, 1.51957436358475e-33)
(50.26548245743669, 1.51957436358475e-33)
(94.2477796076938, 1.32563038769384e-33)
(64.38711959055742, 0.000241155549725919)
(-59.69026041820607, 4.21786179228739e-34)
(12.566370614359172, 1.51957436358475e-33)
(-64.38711959055742, 0.000241155549725919)
(23.519452498689006, 0.00180451785856468)
(-23.519452498689006, 0.00180451785856468)
(86.38222203472871, 0.000133996378076552)
(-939.3362034233481, 1.82874873069053e-33)
(-51.81698248727967, 0.000372300864235917)
(34.55751918948773, 4.07351440822617e-33)
(15.707963267948966, 1.51957436358475e-33)
(87.96459430051421, 1.51957436358475e-33)
(-14.066193912831473, 0.00502871873123234)
(-9.42477796076938, 1.51957436358475e-33)
(4.493409457909064, 0.0471904492258113)
(20.37130295928756, 0.00240390403096148)
(-29.81159879089296, 0.00112393467820302)
(89.52422093041719, 0.000124756940214054)
(70.6716857116195, 0.000200180676620011)
(-31.41592653589793, 1.51957436358475e-33)
(7.725251836937707, 0.0164800259929739)
Intervalos de crecimiento y decrecimiento de la función:Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
$$x_{1} = 65.9734457253857$$
$$x_{2} = 69.1150383789755$$
$$x_{3} = -21.9911485751286$$
$$x_{4} = 21.9911485751286$$
$$x_{5} = -3.14159265358979$$
$$x_{6} = -15.707963267949$$
$$x_{7} = 6.28318530717959$$
$$x_{8} = 28.2743338823081$$
$$x_{9} = -94.2477796076938$$
$$x_{10} = -12.5663706143592$$
$$x_{11} = -53.4070751110265$$
$$x_{12} = 185.353966561798$$
$$x_{13} = 59.6902604182061$$
$$x_{14} = 56.5486677646163$$
$$x_{15} = 72.2566310325652$$
$$x_{16} = -50.2654824574367$$
$$x_{17} = 78.5398163397448$$
$$x_{18} = -87.9645943005142$$
$$x_{19} = 37.6991118430775$$
$$x_{20} = -6.28318530717959$$
$$x_{21} = -37.6991118430775$$
$$x_{22} = -43.9822971502571$$
$$x_{23} = 47.1238898038469$$
$$x_{24} = -100.530964914873$$
$$x_{25} = 3.14159265358979$$
$$x_{26} = -72.2566310325652$$
$$x_{27} = -81.6814089933346$$
$$x_{28} = -65.9734457253857$$
$$x_{29} = -28.2743338823081$$
$$x_{30} = -56.5486677646163$$
$$x_{31} = -207.345115136926$$
$$x_{32} = 43.9822971502571$$
$$x_{33} = 100.530964914873$$
$$x_{34} = -97.3893722612836$$
$$x_{35} = 81.6814089933346$$
$$x_{36} = -75.398223686155$$
$$x_{37} = 50.2654824574367$$
$$x_{38} = 94.2477796076938$$
$$x_{39} = -59.6902604182061$$
$$x_{40} = 12.5663706143592$$
$$x_{41} = -939.336203423348$$
$$x_{42} = 34.5575191894877$$
$$x_{43} = 15.707963267949$$
$$x_{44} = 87.9645943005142$$
$$x_{45} = -9.42477796076938$$
$$x_{46} = -31.4159265358979$$
Puntos máximos de la función:
$$x_{46} = 45.5311340139913$$
$$x_{46} = 14.0661939128315$$
$$x_{46} = 58.1022547544956$$
$$x_{46} = -11295.596297452$$
$$x_{46} = -20.3713029592876$$
$$x_{46} = -86.3822220347287$$
$$x_{46} = -89.5242209304172$$
$$x_{46} = -45.5311340139913$$
$$x_{46} = -80.0981286289451$$
$$x_{46} = 95.8081387868617$$
$$x_{46} = 73.8138806006806$$
$$x_{46} = -58.1022547544956$$
$$x_{46} = -42.3879135681319$$
$$x_{46} = 29.811598790893$$
$$x_{46} = -73.8138806006806$$
$$x_{46} = 51.8169824872797$$
$$x_{46} = -67.5294347771441$$
$$x_{46} = 36.1006222443756$$
$$x_{46} = -4.49340945790906$$
$$x_{46} = 42.3879135681319$$
$$x_{46} = -95.8081387868617$$
$$x_{46} = 67.5294347771441$$
$$x_{46} = -36.1006222443756$$
$$x_{46} = 92.6661922776228$$
$$x_{46} = 48.6741442319544$$
$$x_{46} = -7.72525183693771$$
$$x_{46} = 26.6660542588127$$
$$x_{46} = -54.9596782878889$$
$$x_{46} = 80.0981286289451$$
$$x_{46} = -39.2444323611642$$
$$x_{46} = -83.2401924707234$$
$$x_{46} = -61.2447302603744$$
$$x_{46} = 64.3871195905574$$
$$x_{46} = -64.3871195905574$$
$$x_{46} = 23.519452498689$$
$$x_{46} = -23.519452498689$$
$$x_{46} = 86.3822220347287$$
$$x_{46} = -51.8169824872797$$
$$x_{46} = -14.0661939128315$$
$$x_{46} = 4.49340945790906$$
$$x_{46} = 20.3713029592876$$
$$x_{46} = -29.811598790893$$
$$x_{46} = 89.5242209304172$$
$$x_{46} = 70.6716857116195$$
$$x_{46} = 7.72525183693771$$
Decrece en los intervalos
$$\left[185.353966561798, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -939.336203423348\right]$$